# aldi shopping

## aldi shopping

Distribution of Exceedances
Since Peaks over Threshold (POT) focuses on the realizations exceeding a given (high)
threshold and the threshold method uses data more efficiently, we concentrate on peak over
threshold approach.
The Pickands-Balkema-de Haan Theorem
Suppose that X 1 , X 2 ,..., X n are n independent realisations of a random variable X with a
distribution function F ( x ) . Let xF be the finite or infinite right endpoint of the distribution
F. The distribution function of the excesses over certain (high) threshold u is given by
Fu ( x ) = Pr { X − u ≤ x X > u} =

F ( x + u ) − F (u )

1 − F (u )

for 0 ≤ x < xF − u.

The Pickands-Balkema-de Haan theorem states that if the distribution function F ∈ DA ( H ξ )
then there exists a positive measurable function σ (u ) such that
lim

u→x F

sup { F ( x) − Gξ σ

, (u )

u

}

( x) = 0

0 ≤ x < xF − u

and vice versa, where Gξ ,σ (u ) ( x) denotes the Generalised Pareto distribution (see below).

The above theorem states that as the threshold u becomes large, the distribution of the
excesses over the threshold tends to the Generalised Pareto distribution, provided the
underlying distribution F belongs to the domain of attraction of the Generalised Extreme
Value distribution.

Generalized Pareto Distribution (GPD)

The GPD is a two parameter distribution with distribution function

1

−1

ξ
ξx
1
1

+

β
Gξ , β ( x ) = 
1 − exp − x
β


(

)

(

)

where β > 0, and where x ≥ 0 when ξ ≥ 0 and 0 ≤ x ≤ − β

ξ ≠0
ξ =0
ξ when ξ 0 then ξ , β is a reparametrized version of the ordinary
Pareto distribution, which has a long history in actuarial mathematics as a model for large
losses; ξ = 0 corresponds to the exponential distribution and ξ < 0 is known as a Pareto type
II distribution.

The first case is the most relevant for risk management purposes since the GPD is heavytailed when ξ > 0. Whereas the normal distribution has moments...